A series for which probability of infinitely many events is 0.

Question

Given real-valued random variables $X_i$ for $i \geq 0$, I need to find a series of constants $a_i \in \mathbb{R}$ such that: $$P(X_i > a_i \text{ for infinitely many } i) = 0$$

My attempt is to use Borel-Cantelli lemma as follows:

$$P(\limsup_{i\to\infty}{\{X_i > a_i\}}) = 0 \implies \sum_i^\infty P(\{X_i > a_i\}) < \infty$$

And then by Markov inequality:

$$\sum_i^\infty \frac{\mathbb{E}X_i}{a_i} < \infty \implies \sum_i^\infty \frac{1}{a_i} < \infty \implies \lim_{i \to \infty} a_i = \infty$$

Does this reasoning make sense?

Answer 1

The implications you wrote at the end are not useful. You have to specify how you choose $a_i$'s.

Take $a_i=2^{i} E|X_i|$ . Thne $P(X_i >a_i) \leq P(|X_i| >a_i) \leq \frac{E|X_i|} {a_i}=\frac 1 {2^{i}}$ so $\sum P(X_i >a_i) <\infty$ and you are ready to use Borel Cantelli Lemma.